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Efficient computation of $r$-th root in $\mathbb F_q$ has many applications in computational number theory and many other related areas. We present a new $r$-th root formula which generalizes M\"{u}ller's result on square root, and which provides a possible improvement of the Cipolla-Lehmer type algorithms for general case. More precisely, for given $r$-th power $c\in \mathbb F_q$, we show that there exists $\alpha \in \mathbb F_{q^r}$ such that $$Tr\left(\alpha^\frac{(\sum_{i=0}^{r-1}q^i)-r}{r^2}\right)^r=c,$$ where $Tr(\alpha)=\alpha+\alpha^q+\alpha^{q^2}+\cdots +\alpha^{q^{r-1}}$ and $\alpha$ is a root of certain irreducible polynomial of degree $r$ over $\mathbb F_q$.
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참고문헌 (19)
참고문헌 신청
L. Adleman / 1977 / On taking roots in nite elds / 18th Annual Symposium on Foundations of Computer Science :175~178
A. O. L. Atkin / 1992 / Probabilistic primality testing :159~163
D. Bernstein / Faster square root in annoying nite eld
Z. Cao / 2012 / Adleman-Manders-Miller root extraction method revisited, in Information security and cryptology / Lecture Notes in Comput. Sci. 7537:77~85
G. H. Cho / 2015 / New cube root algorithm based on the third order linear recurrence relations in nite elds / Des. Codes Cryptogr 75(3):483~495
A. O. L. Atkin / 1992 / Probabilistic primality testing :159~163
D. Bernstein / Faster square root in annoying nite eld
Z. Cao / 2012 / Adleman-Manders-Miller root extraction method revisited, in Information security and cryptology / Lecture Notes in Comput. Sci. 7537:77~85
G. H. Cho / 2015 / New cube root algorithm based on the third order linear recurrence relations in nite elds / Des. Codes Cryptogr 75(3):483~495
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